Chapter 1, Problem 12PS

(a)

To determine

The domain and range of $f\left(x\right)=\frac{1}{1-x}$

Answer to Problem 12PS

The domain is $\left\{R-\left\{1\right\}\right\}$ and the range is $\left(-\infty ,\infty \right)$

Explanation of Solution

Given information:

Consider the following function,

  $f\left(x\right)=\frac{1}{1-x}$

Find the domain and range of f.

Formula used:

Domain of $f\left(x\right)$ is all real values of x

Calculation:

Domain of $f\left(x\right)$ is all real values of x except x = 1 because at x = 1, the denominator will become 0 and function is not defined at this point.

Hence Domain of the function is $\left\{R-\left\{1\right\}\right\}$

The function can result in every possible value of real numbers.

Hence Range of the function is $\left(-\infty ,\infty \right)$

Conclusion:

The domain is $\left\{R-\left\{1\right\}\right\}$ and the range is $\left(-\infty ,\infty \right)$

(b)

To determine

The domain of the function $f\left(x\right)=\frac{1}{1-x}$

Answer to Problem 12PS

The domain of the function is $\left\{R-\left\{0\right\}\right\}$

Explanation of Solution

Given information:

Consider the following function,

  $f\left(x\right)=\frac{1}{1-x}$

Calculate $f\left(f\left(x\right)\right)$

Formula used:

The domain of $f\left(f\left(x\right)\right)$ is all real values of x

Calculation:

Calculate $f\left(f\left(x\right)\right)$

  $\begin{array}{l}f\left(x\right)=\frac{1}{1-x}\\ f\left(f\left(x\right)\right)=\frac{1}{1-f\left(x\right)}\\ f\left(f\left(x\right)\right)=\frac{1}{1-\frac{1}{1-x}}\\ f\left(f\left(x\right)\right)=\frac{x-1}{x}\end{array}$

Domain of $f\left(f\left(x\right)\right)$ is all real values of x except x = 0 because at x = 0, the denominator will become 0 and function is not defined at this point.

Hence Domain of the function is $\left\{R-\left\{0\right\}\right\}$

Conclusion:

The domain of the function is $\left\{R-\left\{0\right\}\right\}$

(c)

To determine

The value of $f\left(f\left(f\left(x\right)\right)\right)$

Answer to Problem 12PS

The graph of the $f\left(f\left(f\left(x\right)\right)\right)=x$ is a straight line as it is a linear function of x.

Explanation of Solution

Given information:

Consider the following function,

  $f\left(x\right)=\frac{1}{1-x}$

Calculate $f\left(f\left(f\left(x\right)\right)\right)$

Formula used:

The $y=x$ is the straight line that passes through the origin.

Calculation:

Calculate $f\left(f\left(f\left(x\right)\right)\right)$

Substitute x with $f\left(f\left(f\left(x\right)\right)\right)=\frac{x-1}{x}\text{in }f\left(x\right)$

  $\begin{array}{l}f\left(f\left(f\left(x\right)\right)\right)=\frac{1}{1-f\left(f\left(x\right)\right)}\\ =\frac{1}{1-\frac{x-1}{x}}\\ f\left(f\left(f\left(x\right)\right)\right)=x\end{array}$

The graph of the $f\left(f\left(f\left(x\right)\right)\right)=x$ is a straight line as it is a linear function of x.

  

Thus the graph of the above function is a line.

Conclusion:

The graph of the $f\left(f\left(f\left(x\right)\right)\right)=x$ is a straight line as it is a linear function of x.